Trigonometric Functions – Lesson 3

REVIEW:Graph Sine and Cosine

FunctionsWith amplitude and period

changes.INVESTIGATE”

VERTICAL SHIFTObjective: To graph sine and

cosine functions with amplitude, period changes

and vertical shift!

What do trig functions model in real life?

• Sound waves• Ferris Wheel• Music frequencies• EKG’s• Just as we can create linear, exponential

and quadratic models to represent real life data, we can also use regression to determine whether or not a trig function would be a good model to represent the data!

Graphing: What do we know – starting point?

f(x) = sin x and f(x) = cos x

Range & Intercepts:f(x) = sin x and f(x) = cos x what is the shift between sin and cos?

f(x) = sin x & two important ideas

Period

Am

plitudePeriod

Period means how many degrees in one cycle.

Amplitude means the distance from the centre to the

maximum or minimum, OR (max + min) ÷ 2A

mplitude

f(x) = sin x

Period

Period = 360º

Amplitude = 1

How does “b” impact the graph?f(x) = sin x & f(x) = sin 2x

Period = 180º

b = 2What does it do?

f(x) = sin x & f(x) = sin 3x

So b changes the period = 360º ÷ b or

If _____ ____1 it’s hard to get out of the water! If _____ ___ 1 it’s easy to get out without getting slammed by a wave!

Period

= 120º

How does “A” impact the graph?f(x) = sin x & f(x) = -1 sin xIs the y intercept the same?

What changes?

Amplitude = 1

f(x) = sin x & f(x) = -3 sin x

Amplitude = 3

f(x) = sin x & f(x) = A sin x

The A gives the amplitude of the function.

A negative value means the graph goes down – up, not up – down.

A = 4

A = -3

Amplititude = “a”If ______ ____ 1 you get a taller

wave! (Think: Hawaii Waves!)

If ________ ___ 1 you get CT shore waves!

Now we will investigate how k impacts the graph!

f(x) = a sin bx + k

Any conjectures about “K”? Where have we seen “k” before?

What did the “K” do in this function?

f(x) = sin x & f(x) = sin x + 3

f(x) = sin x + 3 & f(x) = sin x – 2

So “k” shifts the curve up and down. We call this vertical shift or vertical displacement.

f(x) = asin bx + k

a = amplitude

Note: It is exactly the same for sine and cosine.

The difference is the where it crosses the y-axis.

b = 360º ÷ period

k = vertical shift

What is the equation of this function?

Amplitude = 2

Period = 120º

Vertical shift = -1

f(x) = 2 sin 3x – 1

so, A = 2

so, B = 3

so, k = -1

What is the equation of this function?

Amplitude = 4,

going down-up

Period = 720º

Vertical shift = 1

f(x) = -4 sin ½x + 1

so, A = -4

so, B = 0.5

so, k = 1

What is the equation of this function?

Amplitude = 2.5

Period = 240º

Vertical shift = 2

f(x) = 2.5 sin 1.5x + 2

so, A = 2.5

so, B = 1.5

so, k = 2

Sketch the graph of y = 2 sinx - 4

A = ________ b = __________ k = _______

Period = _______________

5 critical points:Range:

Sketch the graph of y = sin2x - 4

A = ________ b = __________ k = _______

Period = _______________

5 critical points:

Range:

Put it altogether! Sketch the graph of

a = ________ b = __________ k = _______

Period = _______________

5 critical points

Range

Graph. Then check your graph on your gc!

1. Y = -2cosx + 4 2. y = .5sin2x - 2